Riemann domain, complex (-analytic) manifold over
An analogue of the Riemann surface of an analytic function of a single complex variable for the case of analytic functions , , of several complex variables , .
More precisely, a path-connected Hausdorff space is called an (abstract) Riemann domain if there is a local homeomorphism (a projection) such that for each point there is a neighbourhood that transforms homeomorphically into a polydisc
in the complex space . A Riemann domain is a separable space.
A complex function is called holomorphic on if for any point the function of complex variables is holomorphic in the corresponding polydisc . The projection is given by the choice of holomorphic functions , which correspond to coordinates in . Starting from a given regular element of an analytic function , its Riemann domain is constructed in the same way as the Riemann surface of a given analytic function of one complex variable, i.e. initially by means of analytic continuation one constructs the complete analytic function , and then, using neighbourhoods, one introduces a topology into the set of elements of the complete analytic function. Like Riemann surfaces, Riemann domains arise unavoidably in connection with analytic continuation of a given element of an analytic function when, following the ideas of B. Riemann, one tries to represent the complete analytic function as a single-valued point function on a domain.
In particular, Riemann domains arise as multi-sheeted domains of holomorphy of analytic functions of several complex variables. Oka's theorem states that a Riemann domain is a domain of holomorphy if and only if it is holomorphically convex (see Holomorphically-convex complex space).
Modern studies of Riemann domains are conducted within the framework of the general theory of analytic spaces. A generalization of the concept of a domain of holomorphy leads to Stein spaces (cf. Stein space).
|||B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)|
|||R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)|
|||L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973)|
The notion as presented above of a Riemann domain has been extended in several ways: Instead of one may choose any (model) complex-analytic space (cf. Complex space). An unramified Riemann domain over is a triple where is a complex-analytic space and is a locally biholomorphic mapping from into .
Next, a ramified Riemann domain over is a triple where again is a complex-analytic space and is now a discrete open holomorphic mapping from to [a1].
|[a1]||H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) pp. Chapt. VI (Elraged & Revised Edition. Original: 1934)|
|[a2]||H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) (Translated from German)|
Riemannian domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_domain&oldid=12703